Existence results for a class of nonlocal problems involving p(x)‐Laplacian

This paper is concerned with the existence of solutions to a class of p(x)‐Kirchhoff‐type equations with Dirichlet boundary data as follows: By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

In this paper, we study the initial boundary value problem of nonlinear pseudo‐parabolic equation with a memory term with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (E(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤E(0) < d_k) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Screen‐type boundary value problems for polymetaharmonic equations

In this paper, we consider the three‐dimensional Riquier‐type and Dirichlet‐type screen boundary value problems for the polymetaharmonic equation with real wave numbers k₁ and k₂. We investigate these problems by means of the potential method and the theory of pseudodifferential equations, prove the existence and uniqueness of solutions in Sobolev–Slobodetski spaces, and on the basis of asymptotic analysis, we establish the best Hölder smoothness results for solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.     

On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ? < π ∕ 2 and γ₀ > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ₀ > 0 for given 0 < ? < π ∕ 2. We also prove the maximal L_p ? L_q regularity theorem of the nonstationary Stokes problem as an application of the ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions of periodic‐type boundary value problems for multi‐term fractional differential equations

Results on the existence of solutions of a periodic‐type boundary value problem of singular multi‐term fractional differential equations with the nonlinearity depending on are established and being singular at t = 0 and t = 1. The analysis relies on the well‐known fixed‐point theorems. An example is given to illustrate the efficiency of the main theorems. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of semilinear elliptic equations with prescribed limiting behaviour

In this paper, we consider semilinear elliptic equations of the form Δu + f(u) = 0 over a quarter space with Dirichlet boundary conditions. Given a suitable positive root z of f, we show how to construct a non‐negative bounded solution u converging to a one‐dimensional limiting profile V with V . This is established using Perron's method by constructing sub‐solutions and super‐solutions and employing a sliding argument. Copyright © 2016 John Wiley & Sons, Ltd.     

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

First‐order systems in on with absolutely continuous real symmetric π‐periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet‐type boundary value problems on [0,π] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet‐type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in RN

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities (1.1) where F∈C¹(R^N×R⁺×R⁺,R),V∈C(R^N,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Concentration phenomena for a fractional Schrödinger‐Kirchhoff type equation

In this paper, we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger‐Kirchhoff type equation where ε>0 is a small parameter, is the fractional Laplacian, M is a Kirchhoff function, V is a continuous positive potential, and f is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik‐Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

With the aid of Lenard recursion equations, we derive the Wadati–Konno–Ichikawa hierarchy. Based on the Lax matrix, an algebraic curve of arithmetic genus n is introduced, from which Dubrovin‐type equations and meromorphic function φ are established. The explicit theta function representations of solutions for the entire WKI hierarchy are given according to asymptotic properties of φ and the algebro‐geometric characters of . Copyright © 2015 John Wiley & Sons, Ltd.     

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p ? 1) < 1, r ≤ 1, and . By using L^p‐integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Analytical solution for a generalized space‐time fractional telegraph equation

In this paper, we consider a nonhomogeneous space‐time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the first‐order or second‐order time derivative by the Caputo fractional derivative , α > 0 and the Laplacian operator by the fractional Laplacian ( ? Δ)^{β ∕ 2}, β ∈ (0,2]. We discuss and derive the analytical solutions under nonhomogeneous Dirichlet and Neumann boundary conditions by using the method of separation of variables. The obtained solutions are expressed through multivariate Mittag‐Leffler type functions. Special cases of solutions are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundedness in a quasilinear chemotaxis‐haptotaxis system with logistic source

This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusion subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and (n ?2) is a bounded and smooth domain. Here, we assume that D (u )?c _D u ^{m  ? 1} for all u  > 0 with some c _D  > 0 and m ?1. For the case of non‐degenerate diffusion, if μ  > μ ^?, where it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution u_b by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of u_b is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of u_b as the parameter b↘0. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

In this paper, we consider the elliptic boundary blow‐up problem where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω₊,Ω_? of Ω, respectively, a ₊ vanishes on the boundary of satisfies p (x )≥1 in Ω,p (x ) > 1 on ? Ω and , and ε is a parameter. We show that there exists ε ^?>0 such that no positive solutions exist when ε > ε ^?, while a minimal positive solution u _ε exists for every ε ∈(0,ε ^?). Under the additional hypotheses that is a smooth N ? 1‐dimensional manifold and that a ₊,a _? have a convenient decay near Γ, we show that a second positive solution v _ε exists for every ε ∈(0,ε ^?) if , where N ^?=(N + 2)/(N ? 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a ₊>0 on ? Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions for a modified Kirchhoff‐type equation in RN

In this paper, we study the following modified Kirchhoff‐type equations of the form: where a > 0, b ≥ 0, and . Under appropriate assumptions on V (x) and h(x,u), some existence results for positive solutions, negative solutions, and sequence of high energy solutions are obtained via a perturbation method. Copyright © 2014 John Wiley & Sons, Ltd.